How do I get the children of a frame node?
Instead of looking at the frame node and trying to find its children you have to look at each node and see what its parent is:

import bpy

children = []
nodes = bpy.context.scene.node_tree.nodes
frameNode = nodes['Frame']

for node in nodes:
    if node.parent == frameNode:  
        children.append(node)

print(children)


This creates a list of the frame node's children.

Or the more condensed version:

import bpy

nodes = bpy.context.scene.node_tree.nodes
frameNode = nodes['Frame']
children = [node for node in nodes if node.parent == frameNode]


This is based on compositor nodes so accessing the node tree will be slightly different for materials, but the same logic will apply.